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Math 161, Calculus 1
Sample Final Exam

Name:

Answer all questions.

Give definitions for
1. $\displaystyle{\lim_{x\to a} f(x) =L}$
2. is continuous on
What three major problems give rise to the concept of the derivative?
Show how to find the derivative of $f(x)=\sqrt{x}$ using the definition.
Give a $\delta$ - $\epsilon$ proof that $\displaystyle\lim_{x\to 1} x^3=1$ .
Find the following limits:
1. $\displaystyle{\lim_{x\to 0} \frac{(x^2-1)\sin(x)}{x}}$
2. $\displaystyle{\lim_{x\to 3} \frac{3x^2+2}{\sqrt{x+6}}}$
3. $\displaystyle{\lim_{x\to \infty} \frac{\sqrt{x^2+1}}{2x-1}}$
Find the following derivatives:
1. if $\displaystyle{f(x)=\frac{\tan(x)}{x^2+\frac{1}{x^2}}}$
2. $\frac{dy}{dx}$ if $y= \sin(x+\sqrt[3]{x^2+3x})$
3. if $f(x)=(x^5+3)(\cos(x))\sqrt{x^3+1}$
State carefully the following theorems:
1. The Intermediate Value Theorem
2. The Mean Value Theorem
Give a proof from the definitions of derivative and limit of the following part of the Candidate Theorem:
If then assumes neither a local maximum nor a local minimum at a.
Prove the squeeze theorem.
Prove from the definition that if is a constant.
A jewelry box is to be made with a square top. The box is to have a volume of 1000 cm. What are the most economical dimentions if the material for the top costs three times as much as the material for the sides and bottom?
Consider the function
1. Where is increasing?
2. Where is concave down?
3. What are the asymptotes of ?